The function provided in the image is:

$f(x) = -4x^4 + 2x^2 - 4x + 6$

To list all possible rational zeros using the Rational Zero Theorem, we need to:

1. Identify the constant term ($p$) and the leading coefficient ($q$).

   • Constant term ($p$) = 6
   • Leading coefficient ($q$) = -4

2. Find all factors of $p$ and $q$:

   • Factors of $p = 6$: $\pm 1, \pm 2, \pm 3, \pm 6$
   • Factors of $q = -4$: $\pm 1, \pm 2, \pm 4$

3. List all possible rational zeros using the formula: $\text{Possible rational zeros} = \frac{\text{factors of } p}{\text{factors of } q}$

The possible rational zeros are: $\pm 1, \pm 2, \pm 3, \pm 6, \pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{1}{4}, \pm \frac{3}{4}$

Thus, the possible rational zeros are: $\pm 1, \pm 2, \pm 3, \pm 6, \pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{1}{4}, \pm \frac{3}{4}$

Would you like to check any of these values or have additional questions?

Related Questions:

1. How do we test which of the possible zeros are actual zeros?
2. Can the Rational Zero Theorem give non-rational zeros?
3. What is the difference between real and rational zeros?
4. How does synthetic division help verify possible zeros?
5. What are some limitations of the Rational Zero Theorem?

Tip: It's always useful to graph the function to visualize where the real zeros might be before testing values.